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UNIVERSITI TENAGA NASIONALCOLLEGE OF ENGINEERINGDEPARTMENT OF MECHANICAL ENGINEERINGCOEB2014 - NUMERICAL METHODS FOR ENGINEERS(GROUP PROJECT REPORT)GROUP TEAM NAME : MASMONAGROUP MEMBERS:1. NAME: AMIRUL AZRAI BIN CHE AZMAN ID: ME2. NAME: KAMARUL HAKIMY BIN KAMARUL ID: MEZULKARNAIN3. NAME: MUHAMMAD A...

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UNIVERSITI TENAGA NASIONAL COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING

COEB2014 - NUMERICAL METHODS FOR ENGINEERS (GROUP PROJECT REPORT)

GROUP TEAM NAME : MASMONA GROUP MEMBERS: 1. NAME: AMIRUL AZRAI BIN CHE AZMAN

ID: ME0105481

2. NAME: KAMARUL HAKIMY BIN KAMARUL

ID: ME0105432

ZULKARNAIN 3. NAME: MUHAMMAD AIMAN BIIN MOHD MUTAZAR

ID: ME0105587

4. NAME: MUHAMAD AMIR HAZIQ BIN MUHAMAD

ID:

ME0105486 NAZIR

SECTION

: 2A

INSTRUCTOR

: MADAM KARMILA KAMIL

GROUP

: MASMONA

Submitted Date 14/12/2020

TABLE OF CONTENTS

ABSTRACT............................................................................................................................1 ACKNOWLEDGEMENT.....................................................................................................2 1.0 INTRODUCTION............................................................................................................3 1.1 Objective........................................................................................................................4 1.2 Literature Review.........................................................................................................4 2.0 RESULTS AND DISCUSSION.......................................................................................5 2.1 Question 1......................................................................................................................5 2.1.1 Results and Discussion........................................................................................................5

2.2 Question 2......................................................................................................................7 2.2.1 Result and Discussion.........................................................................................................7

2.3 Question 3......................................................................................................................9 2.3.1. Result and Discussion........................................................................................................9

2.4 Question 4....................................................................................................................13 2.4.1. Result and Discussion......................................................................................................14

3.0 CONCLUSION...............................................................................................................15 4.0 REFERENCE.................................................................................................................16

b

ABSTRACT

Numerical Methods for Engineering has given us the task of the final project. This final project that has been given to us has a specific topic for us this semester (Semester 1 2020/2021). We have been given the T-Junction situation for this current final project. This final project is one of the important phases for student to step up their knowledge, idea and experience. This subject gives the student the experience of a real life situation to train the students on how to solve the problems during the period given to complete this final project. This subject final project can make every student who is involved with the project to evaluate themselves and their teammates the level of their experience, knowledge, idea and work ethic of everyone involved. This subject is perfectly assigned by our lecturer after finishing the theoretical part of the subject. This will make the quality of the work to be better. This project consist group of 4 students which will encourage the students to learn the real meaning of teamwork to ensure this project to work perfectly. We have discussed and found out the solution in order to reach the goal of our project.

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ACKNOWLEDGEMENT

First of all we would like to forward utmost gratitude to our lecturer Madam Karmila who is our lecturer for subject Numerical Methods for Engineers (COEB2014) in this semester and has given us the knowledge of numerical methods that will help us to encounter and solve the problems in this final project. We would like to thank her for helping us and giving us positive opinions and knowledge so that we can achieve our goal for this final project. Special thanks to every team member who has given full attention and endless commitment to make this final project into a successful project.

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1.0 INTRODUCTION

Traffic congestion is a critical problem which happens on roads which make traffic busy because roads full of cars and buses. Traffic congestion challenges traffic flow in urban area and is prevented smooth traffic. A growing urban area creates complex problems in daily life with traffic. Congestion phenomena cannot be terminated only by applying physical constructing such as build bridge, motorways and increasing road capacity. It is necessary to build technology system like traffic light for transportation management which is used for control of the traffic phenomenon. Traffic control systems have direct influence on traffic problems which is helping to improve traffic flow and reduce traffic congestion. Usually, traffic jams are caused by many reasons such as incidents, works in roads, roads maintenance. Approximately, traffic congestion occurs at peak times in the morning or evening when people are travelling to work.

A study has been made on the traffic congestion in a town which is a T-junction area. The results of the average number of vehicles that pass through the 5 routes for 1 month during the peak hour period is recorded. Based on the results obtained, some problems related to traffic congestion need to be solved. We decided to use mathematical software to solve all the four problems. The mathematical software that we plan to use are Excel and Matlab because all the problems are numerical method calculation.

After going through the problems, it can be said that all the solutions are related to each other. As an example of our question project, to solve for problem question 2, we need to obtain the equation that we get in question 1. So, we can’t simply solve other problem without considering the order of the question. Plus, all the problems are topics that we have covered throughout the semester. As an example, Newton Interpolating Polynomial, Numerical Differentiation, False Position and Simplex methods.

Our group of four have decided to pull together to solve this issue so that traffic congestion can be avoided especially during the rush hour(8am-9am) when the data was taken. By solving all the problems, our team hopes that traffic will be better. 3

1.1 Objective

The objective of this project is to study the estimated route with the highest number of vehicles. Furthermore, the study is to determine the rate of change on the average number of vehicles that pass through the T-junction for every route. Moreover, the study is to find the maximum rate of change on the average number of vehicles between 30 minute and 60 minute. Finally, the study is to determine the type of traffic light that TrafficM needs to focus on their production to maximize their profit.

1.2 Literature Review

The purpose of this project is to study the traffic congestion in a town which requires to consider a T-junction in the area. The traffic of the junction is recorded by the number of vehicles that pass through the T-junction for 5 different routes during peak hour at four intervals between 8.00am to 9.00am which is timed at 15, 32, 46 and 60 minutes. A table have been given that records the average number of vehicles that pass through the 5 routes for 1 month during the peak hour period. Mathematical software such as Excel and Matlab are used to solved the problem of traffic congestion for numerical methods calculation. For the first question, equations of 3rd order polynomial is used to estimate the route with highest number of vehicles at time equal to 30 minute. Next, the second question is developed from question 1 to find the rate of change on average number of vehicles that pass through the Tjunction by using step size of 10 minute. The numerical difference of O(h^2) is used for the calculation to get the answer. Furthermore, false position method is used to find the time for routes that has a maximum rate of change on the average number of vehicles between 30min to 60min for question number three. Lastly, linear programming is set-up to determine the type of traffic light that manufacturing company TrafficM needs to focus on their production to maximize their monthly profit. Simplex method is used to solve the optimization problem in the linear programming of the possibility whether the aim to focus on production of 1 type of traffic light is possible to the company in order to achieve maximum profit per month.

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2.0 RESULTS AND DISCUSSION 2.1 Question 1. 1) Based on the data given, estimate the route with highest number of vehicles at t = 30 min. Develop equations of 3rd order polynomial for your calculations.

2.1.1 Results and Discussion. Route 1:

The number of car when t = 30 is 4.8594 which is 4 cars in route 1.

Route 2:

The number of car when t = 30 is 8.7545 which is 9 cars in route 2.

Route 3:

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The number of car when t = 30 is 8.173 which is 8 cars in route 3. Route 4:

The number of car when t = 30 is 5.905 which is 6 cars in route 4.

Route 5:

The number of car when t = 30 is 4.7235 which is 5 cars in route 5. *By calculating the average cars in each route from route 1 to 5 we have found out that route 2 has the highest number of cars which is 9 cars.

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2.2 Question 2 2.2.1 Result and Discussion From the equations developed in Question 1, first need to find equation to determine the value for ƒ(t) and find the rate of change on the number of vehicles pass through all five routes using step size, h = 10 min. Use numerical differentiation of 0(h2) for your calculation and use formula of centred finite divided difference for intermediate data points. Formula used to find the rate of change on the number of vehicles: 

Forward Finite-Divided Difference : ƒ’(x) = {(- ƒ(xi + 2) + 4 ƒ(xi + 1) - 3 ƒ(xi)}/2(10)



Backward Finite-Divided difference : ƒ’(x) = {(3 ƒ(xi) - 4 ƒ(xi – 1) + ƒ(xi – 2)} / 2(10)



Centered Finite-Divided Difference : ƒ’(x) = {( ƒ(xi + 1) – ƒ(xi – 1)} / 2(10)

Route 1:

Route 2:

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Route 3:

Route 4:

ƒ(t)=4 + 0.1176(t - 15) + (-0.0015)(t - 15)(t - 32) + 0.0002(t - 15)(t - 32)(t - 46) = 0.0002t^3 - 0.0201t^2 + 0.7165t - 2.9 = 0.0006t^2 - 0.0402t + 0.7165 By using equation ƒ'(t) and ƒ(t) : t ƒ(t) ƒ'(t)

0 0.7165 -0.0402

10 0.3745 -0.0282

20 0.1525 -0.0162

Route 5

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30 0.0505 -0.0042

40 0.0685 0.0078

50 0.2065 0.0198

60 0.4645 0.0318

2.3 Question 3 3) Find the time for route(s) that has a maximum rate of change on the average number of vehicles between 30 min and 60 min. (Hint: Maximum happen when the first derivative of a function, dy/dt = 0)

2.3.1. Result and Discussion. First Step: Find first derivative of every route function (fx). Route 1:

Route 2:

Route 3:

9

Route 4:

Route 5:

Second Step: Using False Position Method i) Using those formula to find error to be used in the bracket.

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ii) Then, apply the formula below in order to use the false position method to solve the question.

Route 1:

Route 2:

11

Route 3:

12

Route 4:

Route 5:

2.4 Question 4

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A new traffic light manufacturing company, TrafficM plan to focus their production to one type of traffic light only in order to achieve the best profit per month by considering the constraints related to the production. TrafficM firstly consider three types of traffic light which are time-based light, queue-based light and artificial intelligence-based (AI-based) light. Each type of traffic light requires development’s cost and different production times. They are also sold in the market at different prices as stated in the following table:

The company is only able to provide a capital cost of RM 500 thousand for development cost per month. Due to working hours restrictions, the plant is only able to operate for 21 days per month. Since the company is new, the company has limited storage size which can store only 10 sets of traffic lights in a month. Other monthly costs required to produce the products amount to RM70000. (a) Based on the information given above, set-up the linear programming to determine the type of traffic light that TrafficM needs to focus on their production to maximize their monthly profit. (b) Using the Simplex method, solve the optimization problem in part (a). Discuss the possibility whether the aim to focus on production of 1 type of traffic light is possible to the company in order to achieve maximum profit per month.

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2.4.1. Result and Discussion. a) Linear programming

b)

Using Simplex Table

3.0 CONCLUSION

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As for this final project for Numerical Methods for Engineers, we may conclude that the methods that we learn theoretically from the classes has been very useful for us to finish this final project and the question given. We have been given multiple tasks for this final project that we need to address. The situation given was the T-Junction situation. First of all, we need to find the highest average vehicles among the five routes provided. We used the Newton’s Interpolating Polynomial method to find which route has the highest number of cars. Then for question 2 we used the method that we have studied in Chapter 7 to find the rate of change on the numbers of cars. Moreover, we also used the False Position method to solve the question 3 that require us to find the maximum rate of change on the average number of vehicles between 30 min. and 60 min. This is being used for the application in question 3 and we manage to acquire the answer after we used the first derivative of the function taken from question 1. Lastly, for question 4, we used the simple method that we learn for the Simplex method that provides us the easiest way to acquire the answer we needed. This is a prove that the syllabus of Numerical Methods for Engineers were very useful for us to apply it in the real world to help us to find out the solutions to improve and solve the problems. Also, we have achieve our main objective for this project.

4.0 REFERENCE

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1. S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 7th Edition (2015), Mc.Graw-Hill Inc. Retrieved from: https://www.academia.edu/31766935/Numerical_Methods_for_Engineers_Sixth_Editi on Online, [Accessed, 3 December, 2020]. 2. https://www.youtube.com/watch?v=amVLQSVUDa0 3. https://www.youtube.com/watch?v=Jqa-aFE9-GI 4. https://www.facebook.com/groups/387639758753916

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